Re: Density of a Black Hole?
carlip-nospam_at_physics.ucdavis.edu
Date: 02/11/05
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Date: Fri, 11 Feb 2005 20:48:17 +0000 (UTC)
David Park <djmp@earthlink.net> wrote:
> Can a black hole be said to have a density? Is there a well defined volume
> for a black hole?
No. To define a volume, you need to specify the "time slice" -- the
spacelike hypersurface -- upon which you are looking at the volume.
Outside the event horizon, a black hole is stationary, so there is
a "preferred" choice of time. But inside the horizon, the geometry
is time-dependent, and there is at least no obvious way to specify
a time at which you are determining the volume.
> Can one just integrate the r coordinate from 0 to the Schwarzschild
> radius?
No. Inside the horizon, r is timelike, not spacelike. (Look at the
metric. Inside the horizon, it's the dr^2 term that comes with a
sign different from the others.) If you want to use Schwarzschild
coordinates -- not obviously the "right" thing to do in the interior,
but it's *a* choice -- you should integrate with respect to t at
fixed r.
> In the Foster & Nightingale text: 'A Short Course in General Relativity'
> they have a problem asking to compare the densities of two objects
> that have just become black holes.
It's a misleading question. It makes no real sense. Apart from everything
else wrong with it, you shouldn't use the Schwarzschild metric to answer
the question -- if you are talking about the density of matter, you need
to solve the Einstein field equations with matter present, not the vacuum
equations. There are many possible solutions, which depend on the details
of the matter (the equation of state).
Steve Carlip
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